With the help of the continuation theorem in coincidence degree theory, we establish the existence of positive periodic solutions of impulsive predator-prey system with Holling III functional response.

The paper generalizes a result of R. Bruck and makes an example which shows that the generalization is precise.

Let $M^n$ be a compact submanifold of $S^{n+p}(c)$ with constant scalar curvature. In this paper, we prove that if the squared norm $S$ of the second fundamental form satisfies a certain inequality, then $M^n$ is a totally umbilic or eqality holds and we described all $M^n$ that satisfy this equality.

The aim of this work is to present a classification of some compact hypersurfaces $M^{n}$ of a unit sphere $S^{n+1}$ provided the spectra of the Laplacian of $p$-forms of $M^{n}$, which we denote by $\mathrm{Spec}^p(M)$, is equal to the spectra $\mathrm{Spec}^p(M_0)$, of a given hypersurface $M_{0}^{n}$.

We show that finiteness of a projective volume implies finiteness of total curvature for stochastic complete minimal surfaces with finite number of ends and finite genus which may not be geodesically complete. The tools we use include simple stochastic calculus and Nevanlinna theoretic method.

We present a computational method for obtaining generic forms of sextics with a given configuration of local singularities. Using this method, we first complete the classification of topological types of local singularities appearing on reduced sextics. Next we give the list of possible configurations of singularities containing at least one non-simple singularity and satisfying $\rho(5)\ge 7$, and show that such a sextic is of torus type, which has been conjectured by the third author.

In this paper, we prove that if the derivatives of two nonconstant meromorphic functions $f$ and $g$ share three small functions CM$^*$, or share two small functions CM$^*$ and another two small functions IM$^*$, then $f'=g'$ or $f'$ is a quasi-M\"{o}bius transformation of $g'$ mostly.

In this paper,we study uniqueness of meromorphic functions that share one small function CM with their derivatives. We mainly obtain a uniqueness theorem which answers the questions provided by Kit-wing Yu.

A glued Riemannian space is obtained from Riemannian manifolds $M_1$ and $M_2$ by identifying their isometric submanifolds $B_1$ and $B_2$. A curve on a glued Riemannian space which is a geodesic on each Riemannian manifold and satisfies certain passage law on the identified submanifold $B:=B_1 \cong B_2$ is called a $B$-geodesic. Considering the variational problem with respect to arclength $L$ of piecewise smooth curves through $B$, a critical point of $L$ is a $B$-geodesic. A $B$-Jacobi field is a Jacobi field on each Riemannian manifold and satisfies certain passage condition on $B$. In this paper, we extend the Morse index theorem for geodesics in Riemannian manifolds to the case of a glued Riemannian space.

By using adjunction theory, we describe the smooth complex projective threefolds admitting a simple normal crossing divisor of the form $A+B$ among their hyperplane sections, both components $A$ and $B$ being smooth surfaces with sectional genus $0$, and one of them being nef or at worst an exceptional divisor of the first reduction mapping.

Using the Newton algorithm we show how to compute all the polar quotients and their multiplicities of a plane curve $f=0$, where $f$ is a formal power series of two variables over an algebraically closed field $\bk$ with characteristic zero. The curve is not necessarily reduced.

If $B$ is a Blaschke product with zeros $\{a_n\}$ and if $\sum_n(1-|a_n|)^{\alpha}$ is finite for some $\alpha \in (1/2,1]$, then limits are found on the rate of growth of $\int_0^{2\pi} |B'(re^{it}|^p\, dt$ in agreement with a known result for $\alpha \in (0,1/2)$. Also, a converse is established in the case of an interpolating Blaschke product, whenever $0<\alpha<1$.